Communication cost of consensus for nodes with limited memory

Significance Algorithms that allow a large number, n, of processors to reach consensus are of substantial current interest due to applications in sensor networks and blockchains. When each processor is assigned an initial bit, the consensus bit should match the majority of these bits with high probability. We present a consensus algorithm where the total number of communications between all processors grows linearly in n, yet each processor uses surprisingly few bits of memory; we also prove a lower bound that shows that this memory requirement is sharp up to a factor of three. Our result contrasts with previous algorithms where the consensus matches the majority with probability one at the cost of a superlinear number of communications. Motivated by applications in wireless networks and the Internet of Things, we consider a model of n nodes trying to reach consensus with high probability on their majority bit. Each node i is assigned a bit at time 0 and is a finite automaton with m bits of memory (i.e., 2m states) and a Poisson clock. When the clock of i rings, i can choose to communicate and is then matched to a uniformly chosen node j. The nodes j and i may update their states based on the state of the other node. Previous work has focused on minimizing the time to consensus and the probability of error, while our goal is minimizing the number of communications. We show that, when m>3⁡log⁡log⁡log(n), consensus can be reached with linear communication cost, but this is impossible if m<log⁡log⁡log(n). A key step is to distinguish when nodes can become aware of knowing the majority bit and stop communicating. We show that this is impossible if their memory is too low.